序半群的K-理论

ALGEBRAIC K-THEORY METHOD ON PARTIALLY ORDERED SEMIGROUPS

设S为幺半群,1为其单位元,B是非空集合.若有映射(S在B上的作用)S×B→B满足s(tb)=(st)b,1b=b,其中s,t∈S,b∈B,则称B为(左)S-系.宋光天利用有限生成投射S-系讨论了半群的Grothendieck群和Whitehead群.在文[6]中,作者给出了无零元序幺半群S上的投射序S-系的结构.本文首先利用不可分强凸子系给出了序S-系的分解定理,然后给出了投射序S-系的结构,最后讨论了序半群上的Grothendieck群.

For a monoid S, a （left） S-act is a non-empty set B together with a mappingS × B → B sending （s, b） to sb such that s（tb） = （st）b and 1b = b for all s, t ∈ S andb ∈ B. Using the category of finitely generated projective S-acts, Song introducedthe Grothendieck groups and the Whitehead groups of semigroups. Partially orderedacts over a partially ordered monoid S, or S-posets, appear naturally in the studyof mappings between posets. Recently, projective S-posets without zero element areconsidered. In this paper, a unique decomposition theorem of S-posets is given in termsof strongly convex, indecomposable S-subposets, and a structure theorem for projectiveS-posets is given. In the last section, we discuss the Grothendieck groups of partiallyordered semigroups.